Philosophy from the mind of a fighter

Archive for the category “Philosophy”

In my previous post introducing the concept of Set Theory, we discussed one method for constructing the Natural numbers– a method often referred to as a Von Neumann construction. Using that method, we start with the Empty Set () and then systematically build the Natural numbers by following a rule. As described in that post, this was a step-wise process: look at a number, find its successor, look at the new number, find its successor, repeat ad infinitum. Now, obviously, given a finite amount of time there would be no way to perform this process enough times to generate every Natural number, since every new number we create would still have yet another number succeeding it.

But what if we want to discuss the whole set of Natural numbers?

As we just noted, we cannot construct the Natural numbers in a step-wise manner in order to get all of them. However, mathematicians like Ernst Zermelo, Abraham Fraenkel, and Thoralf Skolem devised a very clever way to take the very same ideas from our step-wise construction in order to discuss a whole, completed set. We refer to this notion as the Axiom of Infinity, and it is one of the premises which underlies the vast majority of modern mathematics.

Infinity minus an infinity yields logically impossible scenarios. Notably, one can take away identical quantities from identical quantities and arrive at contradictory remainders.

On the face of it, this claim appeals to our intuitive understanding of subtraction. If I were to claim that there exists some Integer, , such that and , then we stumble upon the contradiction that . Subtracting identical quantities from identical quantities should yield identical results.

In general, I attempt to avoid topics of morality in my discussions of theology and religion. This is not because I think the topic is too difficult or too complex to discuss, but rather because it seems like an exercise in futility to argue about morality when the foundational metaphysical views upon which we frame morality differ so greatly. I’ve found that such discussions very frequently either end up with one person expounding on the virtues of apples while the other extols the dangers of oranges; or else they end up regressing backward until we are discussing the foundational metaphysics instead of the moral topic.

However, today, I’m going to break that trend. I was recently directed to an article written in November of 2017 entitled Nine Points about Biblical Slavery and Skeptics’ Condemnation of the Bible. The author of the article intends the piece to be a defense of the Bible against complaints regarding the manner in which its constituent documents treat the institution of slavery. The topic is an understandably sore one for a great number of people. Opponents of religion are quite often quick to point out that the Ten Commandments, a list of some of the supposed most important tenets of morality, includes things like, “Keep the Sabbath holy” and “Honor thy father and mother,” but omits something like, “Thou shalt not own another person as your property.” In fact, the Bible never offers any clear proscription against slavery, and indeed seems to sanction the practice in a number of places. Given the incredibly strong modern moral aversion to the idea of slavery, Christian apologists have attempted to take up the task of showing that these modern detractors hold deep misunderstandings of what the Bible actually says about slavery.

The reason I’m breaking my usual avoidance of topics of morality to discuss this one is that I honestly don’t care about the metaphysical foundations of our ethics, in regards to slavery. All I care about, regarding this topic, are the answers to two, simple questions.

Is it morally acceptable to own another human being as property? Does the Bible sanction the owning of another human being as property?

On a blog called Theolocast, Christian apologist Todd Clay recently published an article entitled “31 Reasons to Believe in the God of the Bible.” In the article, Mr. Clay discusses a plethora of different ideas by which he claims that “the God of the Bible has made himself obvious to the world.”

Despite Todd Clay’s assertions, God’s existence is still not obvious to me. In fact, the arguments which he presents are quite bad. Indeed, it seems to me that he is claiming to have found Jesus in his philosophical toast.

The vast majority of people never think about what they mean when they use numbers to describe things. The concept is so ingrained into our early development that we simply take for granted the fact that people understand us when we apply numbers to different things. For most people, numbers are simply numbers, and questions about the meaning of those numbers are confusing and seemingly nonsensical.

Mathematicians are not most people.

For quite a long time, now, mathematicians have recognized that there are at least two very distinct ways in which we use numbers to describe things. Being the scholarly, academic types that they are, mathematicians have assigned names to these two different types of numbers which sound heady and difficult to the average person: ordinal numbers and cardinal numbers. Indeed, even mathematics students sometimes need quite a bit of work and explanation in order to really grasp the difference between these two types of number; but I’m going to do my best to explain these things in a very simple way for a casual audience.

Ordinal and cardinal numbers roughly correspond to the ideas of value and size, respectively.

In his seminal work, Summa Theologica, the celebrated Christian philosopher, Thomas Aquinas, engages with the question of the existence of God. He notes that there are certainly objections to the claim that there exists such a divinity, but Aquinas believes that these objections can be overcome and that this existence can be shown to be well-founded. The eminent philosopher then lays out a list of arguments which he supposes to make this case. These arguments have come to be known as Aquinas’ Five Ways, and they have been so influential in philosophy that many theologians and apologists still cite them as if they are authoritative logical proofs, more than 700 years after the Italian priest set them to page. On the contrary, however, it seems that there are a number of issues which prevent Aquinas’ Five Ways from being quite so powerful, now, as they may have been in his own day.

Joseph Nebus has recently written a couple of posts (here and here) in which he discusses an interesting attempt by Józef Maria Hoëne-Wronski to create a purely numerical definition of the mathematical constant π which is independent of the classical, geometric definition of “the ratio of the circumference of a circle to its diameter.” This has been a goal of many mathematicians, since the idea of π seems like it is more fundamental to mathematics than a definition based on circles would make it seem– as evidenced by the fact that it shows up in areas of mathematics which are seemingly unrelated to circles. Wronski’s idea, to this end, was the following formula:

At first glance, the formula seems inherently nonsensical. After all, is not a number, and therefore cannot be utilized in numerical operations in this way. However, one can get a sense of what Wronski may have intended by this equation. It appears that Wronski wanted to utilize to represent an infinite number, and modern mathematics actually gives us several tools for handling this sort of idea. One which might be of particular use, here, is Non-Standard Analysis with its infinite and infinitesimal Hyperreal numbers. In NSA, we have the ability to perform calculations with and upon infinite numbers perfectly consistently and reasonably.

One of the common claims which is utilized in arguments for the existence of God is that actual infinities cannot exist, implying that there cannot be an infinite regress of causal events in the history of the universe. If there cannot be such an infinite regress, then there must be some First Cause. Theologians then put forth other arguments attempting to show that this First Cause must be God. Blake Giunta, a Christian apologist, has constructed a very interesting and quite useful website cataloging common lines of argumentation from both sides of the debate (color coded Green for Christian arguments and Red for opposing arguments), along with citations and documentation for those claims, called BeliefMap.org. It does not take very long for a fairly cursory perusal of Belief Map to bring one to this exact claim regarding the actually infinite.

While I disagree with Mr. Giunta on many of his views, I have a great deal of respect for him and I think that his work with Belief Map is absolutely fantastic. He truly does attempt to give an irenic and charitable view to the positions of his opposition, and he does sincerely want to discuss the actual arguments being made, instead of being content to knock down Straw Men. To that end, I would like to help Mr. Giunta add to his encyclopedia of apologetics by addressing the manner in which one might answer the claims about actual infinities.

In the video, Dr. Wildberger claims that there are three different ways in which is commonly discussed: the Applied, the Algebraic, and the Analytical. He does a fairly good job of discussing the manner in which the ancient Greeks discovered that there exists no ratio of two whole numbers which can be equal to , which is a topic I have covered here, as well. He then explains what he means by each of the above three categories.

The mathematics of change are quite interesting. In a naive sense, we can often describe a change by a simple collection of data points. For example, let’s think about a little boy rolling a ball across the floor. The boy pushes the ball, and four seconds later, the ball has come to be 2 meters away from him. Given these data points, we may attempt to connect them in some meaningful analytical manner– perhaps by saying that the ball rolled at a speed of half a meter per second. But even this is a somewhat naive bit of information, as it only really tells us something about the completed journey. Mathematicians are greedy, however; they want to be able to know about every point of the ball’s travel, at any arbitrary moment in time.

We can use a function for just such a purpose. A function is a specific mathematical tool which allows us to describe an entire set of data points all at once which we symbolize as (read “ of “). We encode the data by means of a mathematical formula. For example, our exemplary rolling ball might well have been encoded by the function , where the represents the time, in seconds, that the ball has been rolling, and the value of the function, tells us the distance in meters which the ball has traveled in that time. In this particular function, the coefficient of tells us the rate at which distance changes as time passes– that is, a meter per second. When the boy first rolls it, the ball is traveling at a meter per second; when it finishes it had been traveling at a meter per second; and at any single point during the journey the ball is traveling at a meter per second.

However, this is a very simple example. It describes a situation involving a constant velocity. Things become a bit more muddied when the rate at which a change occurs is, itself, changing.